Let r and h be the radius and height of the cylinder. Then,
A = 2πrh + 2πr2 (Given)
⇒ h =

A−2πr2

2πr

Now, Volume (V) = πr2h
⇒ V = πr2(

A−2πr2

2πr

) =

1

2

(Ar−2πr3)
⇒

dV

dr

=

1

2

(A−6πr2) ... (1)
⇒

d2V

dr2

=

1

2

−12πr ... (2)

Now,

dV

dr

= 0 ⇒

1

2

(A−6πr2) = 0
⇒ r2 =

A

6π

⇒ r = √

A

6π

Now, |

dV2

dr2

|r=√

A

6π

=

1

2

(−12π√

A

6π

) < 0
Therefore, Volume is maximum at r=√

A

6π

⇒ r2 =

A

6π

⇒ 6πr2 = A
⇒ 6πr2 = 2πrh + 2πr2
⇒ 4πr2 = 2πrh ⇒ 2r = h
Hence, the volume is maximum if its height is equal to its diameter.